1Lyapunov Functions in Piecewise Linear Systems: From Fixed Point to Limit Cycle

This paper provides a first example of constructing Lyapunov functions in a class of piecewise linear systems with limit cycles. The method of construction helps analyze and control complex oscillating systems through novel geometric means. Special attention is stressed upon a problem not formerly solved: to impose consistent boundary conditions on the Lyapunov function in each linear region. By successfully solving the problem, the authors construct continuous Lyapunov functions in the whole state space. It is further demonstrated that the Lyapunov functions constructed explain for the different bifurcations leading to the emergence of limit cycle oscillation. Index Terms Lyapunov function, global analysis on piecewise linear systems, stability of nonlinear systems. Piecewise linear systems (PLS), as a kind of hybrid systems, have attracted wide interest. Much study has been done on this class of systems to understand the complex behaviors of the nonlinear systems and make control possible. On one hand, a wide variety of natural and technological systems are frequently modeled in PLS, such as neural oscillators [1], [2]